Hi Abbie,
Being a UK citizen who also has a GRC I understand how you must be frustrated. They make them hard to get and most don't bother from my understanding.
I am a bit confused about the refusal of a visa because you can marry or have a civil union in the UK with other trans people. However when you go to Belgium I would anticipate that you will be asked to produce your passport and UK/EU Driving licence. Most countries don't need a birth certificate due to your own country requiring it as proof when issuing a passport.
If you want a chat about it more privately just PM me but whatever way i wish you all the best in Belgium and your new life
Hugs
Sorcha